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234 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
FIGURE 6.15 Frequency sampling structure for M =4
where L = M−1
2
for M odd, L = M 2 − 1 for M even, and {H k (z) ,k=1,
...,L} are 2nd-order sections given by
H k (z) = cos [̸ H (k)] − z −1 cos [̸ ]
H (k) − 2πk
M
1 − 2z −1 cos ( )
2πk
M + z
−2
(6.14)
Note that the DFT samples H (0) and H (M/2) are real-valued and that
the third term on the right-hand side of (6.13) is absent if M is odd. Using
(6.13) and (6.14), we show a frequency sampling structure in Figure 6.16
for M =4containing real coefficients.
FIGURE 6.16
Frequency sampling structure for M =4with real coefficients
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